This is an announcement for the paper "Isoperimetry of waists and local versus global asymptotic convex geometries" by Roman Vershynin.

Abstract: Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If K has one nicely bounded section, then most sections of K are nicely bounded". This 'existence implies randomness' consequence was proved independently in [Giannopoulos, Milman and Tsolomitis]. The main result represents a new connection between thelocal asymptotic convex geometry (study of sections of K) and the global asymptotic convex geometry (study K as a whole). The method relies on the new 'isoperimetry of waists' on the sphere due to Gromov.

Archive classification: Functional Analysis

Mathematics Subject Classification: 52A20,46B07

The source file(s), localglobal.tex: 28490 bytes, is(are) stored in gzipped form as 0404500.gz with size 9kb. The corresponding postcript file has gzipped size 52kb.

Submitted from: vershynin@math.ucdavis.edu

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http://arXiv.org/abs/math.FA/0404500

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